Phillip Griffiths
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Phillip Griffiths initiated with his collaborators the theory of variation of Hodge structure, which has come to play a central role in many aspects of algebraic geometry and its uses in modern theoretical physics. In addition to algebraic geometry, he has made contributions to differential and integral geometry, geometric function theory, and the geometry of partial differential equations. A former Director of the Institute (1991–2003), Griffiths chaired the Science Initiative Group, which fosters science in the developing world through programs such as the Carnegie–IAS African Regional Initiative in Science and Education.
Click here for his curriculum vitae and here for his bibliography.
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Browsing Phillip Griffiths by Type "Lecture/speech"
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- Atypical Hodge loci(2023-06)Phillip GriffithsTalk based on the paper [BKU] and related works given in the references in that work, and on extensive discussions with Mark Green and Colleen Robles.
54 58 - Curvature properties of Hodge bundles(2018-09-12)Griffiths, PhillipIn algebraic geometry the use of complex analytic methods differential geometry and PDEs | is long standing and far reaching for rather deep reasons the results these methods give (Hodge theory, vanishing theorems) have not been replaced by purely algebraic techniques. Two central areas in the subject are moduli and classification of varieties today we will discuss two results in which the use of analytic methods through the curvature properties of the Hodge bundles plays a central role.
244 239 - Extended Period MappingsGriffiths, PhillipThis lecture will discuss the global structure of period mappings (variation of Hodge structure) defined over complete, 2-dimensional algebraic varieties. Some applications to moduli of general type algebraic surfaces will also be presented.
155 85 - Extended Period MappingsGriffiths, PhillipClay Lecture at the INI, June 2022. The lecture is based on joint work with Mark Green and Colleen Robles. Theme of the lecture is global properties of period mappings with applications to the geometry of completions of moduli spaces.
136 86 - Global properties of period mappings on the boundary(2020)
;Griffiths, Phillip ;Green, MarkRobles, ColleenOutline I. Introduction II. Extension data for a mixed Hodge structure III. Extension data for limiting mixed Hodge structures IV. Period mappings to extension data (A) V. Period mappings to extension data (B) Appendix to Sections IV and V: Examples VI. Local Torelli conditions VII. Global structure of period mappings from complete surfaces324 132 - Hodge Theory and Moduli(2023-04-07)Phillip GriffithsHodge theory provides a major tool for the study of moduli. Conversely, moduli have furnished a significant stimulus for the developement of Hodge theory.
89 100 - Hodge Theory and Moduli(2020)Griffiths, PhillipClay Lecture, based on joint work with Mark Green, Radu Laza, and Colleen Robles. Some of the lecture draws work of and discussions with Marco Fanciosi, Rita Pardini, and Sonke Rollenske.
304 110 - Hodge Theory and Moduli(2018-05)Griffiths, PhillipThis talk will be at the interface of the two topics - moduli and singularities and Hodge theory and degenerations of Hodge structures.
221 234 - Hodge theory and ModuliGriffiths, PhillipThe theory of moduli is an important and active area in algebraic geometry. For varieties of general type the existence of a moduli space $\mathcal{M}$ with a canonical completion $\bar{\mathcal{M}}$ has been proved by Koll\'ar/Shepard-Barron/Alexeev. Aside from the classical case of algebraic curves, very little is known about the structure of $\mathcal{M}$, especially it's boundary $\overline{\mathcal{M}} \diagdown\mathcal{M}$. The period mapping from Hodge theory provides a tool for studying these issues. In this talk, we will discuss some aspects of this topic with emphasis on I-surfaces, which provide one of the first examples where the theory has been worked out in some detail. Particular notice will me made of how the extension data in the limiting mixed Hodge structures that arise from singular surfaces on the boundary of moduli may be used to guide the desingularization of that boundary.
372 266 - Hodge Theory and Moduli(2019-05)Griffiths, PhillipAlgebraic geometry is frequently seen as a very interesting and beautiful subject, but one that is also very difficult to get into; this is partly due to its breadth, as traditionally algebra, topology, analysis, differential geometry, Lie theory and more recently combinatorics, logic,. . .are used to study it. I have tried to make these notes accessible to a general audience by illustrating topics with elementary examples and informal geometric and heuristic arguments, and with occasional side comments for experts in the subject.
262 276 - Isolated Hypersurface Singularities(2019-04-20)Griffiths, PhillipLectures given at the University of Miami during April, 2019
218 101 - Lecture Series at the High School of Economics, Moscow(2018-05)Griffiths, PhillipAlgebraic geometry is the study of the geometry of algebraic varieties, defined as the solutions of a system of polynomial equations over a field k. When k = C the earliest deep results in the subject were discovered using analysis, and analytic methods (complex function theory, PDEs and differential geometry) continue to play a central and pioneering role in algebraic geometry. The objective of these talks is to present an informal and illustrative account of some answers to the question in the title.
404 957 - Limits in Hodge TheoryGriffiths, PhillipAlmost all of the deep results in Hodge theory and its applications to algebraic geometry require understanding the limits in a family of Hodge structures. In the literature the proofs of these results frequently use the consequences of the analysis of the singularities acquired in a degenerating family of Hodge structures; that analysis itself is treated as a "black box." In these lectures an attempt will be made to give an informal introduction to the subject of limits of Hodge structures and to explain some of the essential ideas of the proofs. One additional topic not yet in the literature that we will discuss is the geometric interpretation of the extension data in limiting mixed Hodge structures and its use in moduli questions.
186 133 - Moduli and Hodge Theory(2019-04-05)Griffiths, PhillipTalk at UIC (April 5, 2019), and based in part on joint work in progress with Mark Green, Radu Laza and Colleen Robles (GLR). Selected references to works quoted in or related to this talk are given at the end.
239 318 - Period Mapping at Infinity(2020-05-06)Griffiths, PhillipHodge theory provides a basic invariant of complex algebraic varieties. For algebraic families of smooth varieties the global study of the Hodge structure on the cohomology of the varieties (period mapping) is a much studied and rich subject. When one completes a family to include singular varieties the local study of how the Hodge structures degenerate to limiting mixed Hodge structures is also much studied and very rich. However, the global study of the period mapping at infinity has not been similarly developed. This has now been at least partially done and will be the topic of this talk. Sample applications include: new global invariants of limiting mixed Hodge structures; a generic local Torelli assumption implies that moduli spaces are log canonical (not just log general type); and extension data and asymptotics of the Ricci curvature; a proposed construction of the toroidal compactification of the image of period mapping. The key point is that the extension data associated to a limiting mixed Hodge structure has a rich geometric structure and this provides a new tool for the study of families of singular varieties in the boundary of families of smooth varieties.
351 213 - Positivity and Vanishing TheoremsGriffiths, PhillipExistence theorems are a central part of algebraic geometry. These results frequently involve linear problems where positivity assumptions are used to prove existence of solutions by establishing the vanishing of obstructions to that existence. Beginning with Riemann (algebraic curves), Picard (algebraic surfaces) and continuing into more recent times (Lefschetz, Hodge, Kodaira-Spencer and many others since this work) it has come to be understood that the vanishing theorems are intimatedly related to the topology of algebraic varieties. What remains is the case that although the results are about algebraic varieties, analytic tools are needed to establish them. Moreover the property of positivity also appears in other aspects where analytic methods are needed, an example being the proof of the Iitaka conjecture which is central in the classification of algebraic varieties. The purpose of these lectures is to present, sometimes from an historical perspective, some of the principal aspects of the theory.
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